Depth–energy and depth–force relationships in open channel flows II: Analytical findings for power-law cross-sections

In a recent work [Valiani A, Caleffi V. Depth–energy and depth–force relationships in open channel flows: analytical findings. Adv Water Resour 2008;31(3):447–54], the authors analytically inverted the depth–specific energy and depth–total force relationships for flows in open channels with wide rectangular cross-sections. In the present work, the previous results are extended using a simple analytical perturbation technique to the important class of power-law cross-sections, which can often be used to approximate real fluvial geometry. Assuming the river cross-section slightly different from the rectangular form, the exponent of the power-law defining the river cross-section is subsequently small. The expressions for the specific energy and total force as functions of the water depth are considered in dimensionless form. The inversion of these functions begins with the exact analytical solutions for wide rectangular sections. A perturbation analysis is carried out taking the exponent of the power-law as a small parameter. For a known discharge and for each meaningful value of the specific energy, a subcritical and a supercritical depth are analytically determined, expanding the depth in terms of this small parameter up to the second order. Similarly, for each meaningful value of the total force, a subcritical and a supercritical depth are found analytically, also using a second-order expansion of the flow depth. Examples from classical open channel hydraulics show the consistency of these analytical solutions. An error analysis is presented to provide limits of presented solutions and an analytical technique is proposed to further refine the solutions.